1. Two boxes contain between them 65 balls of several different sizes. Each ball is white, black,
red or yellow. If you take any 5 balls of the same colour at least two of them will always be
of the same size (radius). Prove that there are at least 3 balls which lie in the same box have
the same colour and have the same size (radius).
2. For all positive real numbersa, b, cprove that


3. A square sheet of paperABCDis so folded that Bfalls on the mid-point MofCD. Prove
that the crease will divideBCin the ratio 5 : 3.
4. Find the remainder when 2¹⁹⁹⁰
is divided by 1990.
5. Pis any point inside a triangle ABC. The perimeter of the triangle AB+BC+CA= 2s.
Prove that
s < AP+BP+CP <2s.
6. Nis a 50 digit number (in the decimal scale). All digits except the 26th digit (from the left)
are 1. IfNis divisible by 13, find the 26th digit.
7. A censusman on duty visited a house which the lady inmates declined to reveal their individual
ages, but said — “we do not mind giving you the sum of the ages of any two ladies you may
choose”. Thereupon the censusman said — “In that case please give me the sum of the ages of
every possible pair of you”. The gave the sums as follows : 30,33,41,58,66,69. The censusman
took these figures and happily went away. How did he calculate the individual ages of the ladies
from these figures.
8. If the circumcenter and centroid of a triangle coincide, prove that the triangle must be equi-
lateral.